Weighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods Jianxian Qiu School of Mathematical Science Xiamen University jxqiu@xmu.edu.cn http://ccam.xmu.edu.cn/teacher/jxqiu Collaborators: J. Zhu (NUAA), C.-W. Shu (Brown), H. Zhu (NUPT), X. Zhong (MSU), G. Li (Qingdao), Y. Cheng (Baidu Company, Beijing), M. Dumbser (Trento)
Outline Introduction Numerical results Conclusions
Introduction 1 Introduction We consider hyperbolic conservation laws: { u t + f(u) = 0, u(x, 0) = u 0 (x). Hyperbolic conservation laws and convection dominated PDEs play an important role arise in applications, such as gas dynamics, modeling of shallow waters,... There are special difficulties associated with solving these systems both on mathematical and numerical methods, for discontinuous may appear in the solutions for nonlinear equations, even though the initial conditions are smooth enough. 1
Introduction This is why devising robust, accurate and efficient methods for numerically solving these problems is of considerable importance and as expected, has attracted the interest of many researchers and practitioners. Within recent decades, many high-order numerical methods have been developed to solve these problem. Among them, we would like to mention Discontinuous Galerkin (DG) method and Weighted essentially non-oscillatory (WENO) scheme. DG method is a high order finite element method. WENO scheme is finite difference or finite volume scheme. Both DG and WENO are very important numerical methods for the Convection Dominated PDEs. 2
Introduction The first DG was presented by Reed and Hill in 1973, in the framework of neutron transport (steady state linear hyperbolic equations). From 1987, a major development of the DG method was carried out by Cockburn, Shu et al. in a series of papers. They established a framework to easily solve nonlinear time dependent hyperbolic conservation laws using explicit, nonlinearly stable high order Runge- Kutta time discretization and DG discretization in space. These methods are termed RKDG methods. DG employs useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes, and limiters. 3
Introduction Limiter is an important component of RKDG methods for solving convection dominated problems with strong shocks in the solutions, which is applied to detect discontinuities and control spurious oscillations near such discontinuities. Many such limiters have been used in the literature on RKDG methods such as the minmod type TVB limiter by Coukburn and Shu et al., the moment based limiter developed by Flaherty et al.. Limiters have been an extensively studied subject for the DG methods, however it is still a challenge to find limiters which are robust, maintaining high order accuracy in smooth regions including at smooth extrema, and yielding sharp, non-oscillatory discontinuity transitions. 4
Introduction WENO schemes have following advantages: Uniform high order accuracy in smooth regions including at smooth extrema Sharp and essentially non-oscillatory (to the eyes) shock transition. Robust for many physical systems with strong shocks. Especially suitable for simulating solutions containing both discontinuities and complicated smooth solution structure, such as shock interaction with vortices. The limiters used to control spurious oscillations in the presence of strong shocks are less robust than the strategies of WENO finite volume and finite difference methods. In this presentation, we would like to show the design of a robust limiter for the RKDG methods based on WENO methods. 5
2 s We consider one dimensional conservation laws: u t + f(u) x = 0. Let x i are the centers of the cells I i = [x i 1 2, x i+ 1 2 ], x i = x i+ 1 2 x i 1 2, h = sup i x i. The solution and the test function space: V k h = {p : p Ii P k (I i )}. A local orthogonal basis over I i, v (i) 0 (x) = 1, v(i) 1 (x) = x x ( ) 2 i, v (i) x 2 x (x) = xi 1 i x i 12, 6
The numerical solution u h (x, t): u h (x, t) = k l=0 u (l) i (t)v (i) l (x), for x I i The degrees of freedom u (l) i (t) are the moments: u (l) i (t) = 1 u h (x, t)v (i) l (x)dx, l = 0, 1,, k a l I i where a l = I i (v (i) l (x)) 2 dx 7
In order to evolve the degrees of freedom u (l) i (t), we time equation u t +f(u) x = 0 with basis v (i) l (x), and integrate it on cell I i, using integration by part, we obtain: d dt u(l) i (t) + 1 ( f(u h (x, t)) d a l I i f(u h (x i 1/2, t))v (i) l (x i 1/2 ) ) dx v(i) l (x)dx + f(u h (x i+1/2, t))v (i) l (x i+1/2 ) = 0, l = 0, 1,, k However, the boundary terms f(u i+1/2 ) and v i+1/2 etc. are not well defined when u and v are in this space, as they are discontinuous at the cell interfaces. 8
From the conservation and stability (upwinding) considerations, we take A single valued monotone numerical flux to replace f(u i+1/2 ): ˆf i+1/2 = ˆf(u i+1/2, u+ i+1/2 ) where ˆf(u; u) = f(u) (consistency); ˆf(, ) (monotonicity) and ˆf is Lipschitz continuous with respect to both arguments. Values from inside I i for the test function v: v (i) l (x i+1/2 ), v(i) l (x + i 1/2 ) We get semi-discretization scheme: d dt u(l) i (t) + 1 ( f(u h (x, t)) d a l I i dx v(i) l (x)dx + ˆf(u i+1/2, u+ i+1/2 )v(i) l (x i+1/2 ) ˆf(u i 1/2, u i+1/2 )v(i) l (x + i 1/2 ) ) = 0, l = 0, 1,, k. ( ) 9
Using explicit, nonlinearly stable high order Runge-Kutta time discretizations. [Shu and Osher, JCP, 1988] The semidiscrete scheme ( ) is written as: u t = L(u) is discretized in time by a nonlinearly stable Runge-Kutta time discretization, e.g. the third order version. u (1) = u n + tl(u n ) u (2) = 3 4 un + 1 4 u(1) + 1 4 tl(u(1) ) u n+1 = 1 3 un + 2 3 u(2) + 2 3 tl(u(2) ). 10
Lax problem. t = 1.3. 200 cells. Density. Left: k = 1. Right: k = 2. k = 3 code blows up. For Blast Wave problem, code blows up for any k. 11
Limiters Many limiters have been used in the literature, such as: The minmod based TVB limiter.( Cockburn and Shu, Math. Comp. 1989) Moment limiter. (Biswas, Devine and Flaherty, Appl. Numer. Math, 1994) A modification of moment limiter.(burbean, Sagaut and Brunean, JCP, 2001) The monotonicity preserving (MP) limiter. (Suresh and Huynh, JCP, 1997) A modification of the MP limiter. (Rider and Margolin, JCP, 2001) 12
These limiters tend to degrade accuracy when mistakenly used in smooth regions of the solution. Burgers equation, initial condition u(x, 0) = 1 4 + 1 2 sin(π(2x 10)), with periodic boundary condition, RKDG with TVB limiter, t=0.05. Cockburn and Shu, JSC (2001) 13
Burbean, Sagaut and Brunean, JCP, (2001) 14
WENO Type limiter In order to overcome the drawback of these limiters, from 2003, with my colleagues, we have studied using WENO as limiter for RKDG methods, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, non-oscillatory shock transition for RKDG methods. We separate limiter procedure into two parts: Identify the troubled cells, namely those cells which might need the limiting procedure; Reconstruct polynomials in troubled cells using WENO reconstruction which only maintain the original cell averages (conservation). 15
For the first part, we can use the following troubled-cell indicators: TVB: based on the TVB minmod function BDF: moment limiter of Biswas, Devine and Flaherty BSB: modified moment limiter of Burbeau et al. MP: monotonicity-preserving limiter MMP: modified monotonicity-preserving limiter KXRCF: A shock detector of Krivodonova et al., Applied Numer. Math (2004) Harten: Discontinuous detection technique based on Harten s subcell resolution, (Qiu and Shu, SISC, 2005). 16
TVB indicator Let ũ i = u h (x i+1/2 ) u(0) i, ũ i = u h (x + i 1/2 ) + u(0) i. These are modified by the modified minmod function where m is given by ũ (mod) i = m(ũ i, u (0) i+1 u(0) i, u (0) i u (0) i 1 ), ũ (mod) i = m(ũ i, u (0) i+1 u(0) i, u (0) i u (0) i 1 ), m(a 1, a 2,..., a n ) { a1 if a 1 M( x) 2, = m(a 1, a 2,..., a n ) otherwise. 17
The minmod function m is given by m(a 1, a 2,..., a n ) { s min1 j n a = i if sign(a 1 ) = = sign(a n ) = s, 0 otherwise. If ũ (mod) i ũ i or ũ (mod) i ũ i, we declare the cell I i as a troubled cell. 18
KXRCF indicator Partition the boundary of a cell I i into two portions Ii (inflow, v n < 0) and I i + (outflow, v n > 0). The cell I i is identified as a troubled cell, if I (u h i Ii u h Ini )ds > 1, I uh i Ii h k+1 2 i here h i is the radius of the circumscribed circle in the element I i. I ni is the neighbor of I i on the side of Ii and the norm is based on an element average in onedimensional case. 19
WENO reconstruction Reconstruct polynomials in troubled cells using WENO reconstruction which only maintain the original cell averages (conservation). x G is Gauss or Gauss-Lobatto quadrature point 20
S 0 : 1 x S 1 : 1 x S k : 1 x T : 1 x I i+l q 0 (x)dx = u (0) i+l, l = k,, 0; I i+l q 1 (x)dx = u (0) i+l, l = k + 1,, 1; I i+l q k (x)dx = u (0) i+l, l = 0,, k; I i+l Q(x)dx = u (0) i+l, l = k,, k; 21 First Prev Next Last Go Back Full Screen Close Quit
We find the combination coefficients, also called linear weights γ j, j = 0, 1,, k satisfying: A : I i Q(x)v (i) l (x)dx = k γ j j=0 B : Q(x G ) = I i q j (x)v (i) l (x)dx, l = 1,, k k γ j q j (x G ). j=0 22
We compute the smoothness indicator, denoted as β j for each stencil S j, which measures how smooth the function q j (x) on cell I i, β j = k l=1 xi+1/2 x i 1/2 ( x) 2l 1 (q (l) j )2 dx, where q (l) j is the lth-derivative of q j (x). We compute the nonlinear weight ω j based on the smoothness indicator ω j = α j k l=0 α l, with α j = γ j (ε + β j ) 2, j = 0, 1,..., k, where ε > 0 is a small number to avoid the denominator to become 0. 23
The final WENO approximation is then given by: A : u (l) i = 1 a l k ω j j=0 B : u(x G ) = I i q j (x)v (i) l (x)dx, l = 1,, k; k ω j q j (x G ). Reconstruction of moments based on the reconstructed point values: j=0 Remark I: u (l) i = x a l G w G u(x G )v (i) l (x G ), l = 1,, k. For procedure A, there are not the linear weights for P 3 case. 24
For procedure B: For the P 1 case, we use the two-point Gauss quadrature points. For the P 2 case, we use either the four-point Gauss-Lobatto quadrature points or three-point Gauss quadrature points. But there are negative linear weights when three-point Gauss quadrature points are used. For the P 3 case, we use the four-point Gauss quadrature points. Remark II: WENO limiters work well in all our numerical test cases, including 1D, 2D and 3D, structure and unstructured meshes, but for P 2 and P 3 cases, the compactness of DG is destroyed. 25
Hermite WENO (HWENO) reconstruction Reconstruct polynomials which maintain the original cell averages (conservation). 26
For P 2 case, we obtain following reconstructed polynomials: q 0 (x)dx = u (0) i+j a 0, j = 1, 0; q 0 (x)v (i 1) 1 (x)dx = u (1) i 1 a 1 I i+j I i 1 q 1 (x)dx = u (0) i+j a 0, j = 0, 1; q 1 (x)v (i+1) 1 (x)dx = u (1) i+1 a 1 I i+j I i+1 q 2 (x)dx = u (0) i+j a 0, j = 1, 0, 1 I i+j Q(x)dx = u (0) i+j a 0, j = 1, 0, 1; Q(x)v (i+j) 1 (x)dx = u (1) i+j a 1, j = 1, 1. I i+j I i+j Follow the routine A of WENO reconstruction, we can obtain new moment u (1) i. 27
To reconstruct u (2) i : q 0 (x)dx = u (0) i+j a 0, I i+j q 1 (x)dx = u (0) i+j a 0, I i+j q 2 (x)dx = u (0) I i+j I i+j Q(x)dx = u (0) i+j a 0, I i+j q 0 (x)v (i+j) 1 (x)dx = u (1) i+j a 1, j = 1, 0 q 1 (x)v (i+j) 1 (x)dx = u (1) i+j a 1, j = 0, 1 I i+j i+j a 0, j = 1, 0, 1; q 2 (x)v (i) 1 dx = u(1) i a 1 I i I i+j Q(x)v (i+j) 1 (x)dx = u (1) i+j a 1, j = 1, 0, 1, Follow the routine A of WENO reconstruction, we can obtain new moment u (2) i. Remark III: For P 3 case, we should extend stencil. 28
New HWENO type reconstruction I i is a troubled cell, we use stencil S = {I i 1, I i, I i+1 }. Denote the solutions of the DG method on these three cells as polynomials q 0 (x), q 1 (x) and q 2 (x), respectively. We would like to modify q 1 (x) to q new 1 (x). Procedure by Zhong and Shu, JCP (2013): In order to make sure that the reconstructed polynomial maintains the original cell average of q 1 in the target cell I i, the following modifications are taken: q 0 = 1 x i q 0 = q 0 q 0 + q 1, q 1 = q 1, q 2 = q 0 q 2 + q 1 I i q 0 (x)dx, q 1 = 1 x i I i q 1 (x)dx, q 2 = 1 x i I i q 2 (x)dx, 29
The final nonlinear WENO reconstruction polynomial q new 1 (x) is now defined by a convex combination of these modified polynomials: q new 1 (x) = ω 0 q 0 (x) + ω 1 q 1 (x) + ω 2 q 2 (x) If ω 0 + ω 1 + ω 2 = 1, then q new 1 has the same cell average and order of accuracy as q 1. Computational formula of ω 0, ω 1, and ω 2 are same as in WENO reconstruction. The linear weights can be chosen to be any set of positive numbers adding up to one. Since for smooth solutions the central cell is usually the best one, a larger linear weight is put on the central cell than on the neighboring cells, i.e. γ 0 < γ 1 and γ 1 > γ 2. 30
In Zhong and Shu, JCP (2013), they take: γ 0 = 0.001, γ 1 = 0.998 γ 2 = 0.001 which can maintain the original high order in smooth regions and can keep essentially non-oscillatory shock transitions in all their numerical examples. 31
Procedure by Zhu, Zhong, Shu and Q., 2014: In order to make sure that the reconstructed polynomial maintains the original cell average of q 1 in the target cell I i, the following modifications are taken: ( q 0 (x) q 0 (x)) 2 dx = min (φ(x) q 0 (x)) 2 dx I i 1 I i 1 ( q 2 (x) q 2 (x)) 2 dx = min (φ(x) q 2 (x)) 2 dx I i+1 I i+1 for φ(x) P k with I i φ(x)dx = I i q 1 (x)dx For notational consistency we also denote q 1 (x) = q 1 (x). Then we follow the routine of Zhong and Shu JCP (2013), and obtain the final nonlinear WENO reconstruction polynomial q new 1 (x). 32
For two dimensional case, we select the HWENO reconstruction stencil as S = {I i 1,j, I i,j 1, I i+1,j, I i,j+1, I ij } for simplify, we renumber these cells as I l, l = 0,, 4, and denote the DG solutions on these five cells to be p l (x, y), respectively. 33
{ ( p l (x, y) p l (x, y)) 2 dxdy = min (φ(x, y) p l (x, y)) 2 dxdy I l I l ( 2 ( ) } 2 + (φ(x, y) p l (x, y))dxdy) + (φ(x, y) p l (x, y))dxdy, I l for φ(x, y) P k with I 4 φ(x, y)dxdy = I 4 p 4 (x, y)dxdy, where l = mod(l 1, 4) and l = mod(l + 1, 4). For notational consistency we also denote p 4 (x, y) = p 4 (x, y). Then we follow the routine of WENO reconstruction: Take linear weights: γ 0 = γ 1 = γ 2 = γ 3 = 0.001, γ 4 = 0.996 I l 34
We compute the smoothness indicators, β l = k α =1 I ij α 1 where α = (α 1, α 2 ) and α = α 1 + α 2. I ij ( ) α 2 x α p 1 y α l (x, y) dxdy, l = 0,, 4, 2 We compute the non-linear weights based on the smoothness indicators. The final nonlinear HWENO reconstruction polynomial p new 4 (x, y) is defined by a convex combination of the (modified) polynomials in the stencil: p new p new 4 (x, y) = 4 ω l p l (x, y). l=0 4 (x, y) has the same cell average and order of accuracy as the original one p 4 (x, y) on condition that 4 l=0 ω l = 1. 35
Numerical results 3 Numerical results We show the the numerical results of one- and twodimensional cases to illustrate the performance of the WENO type limiters. Accuracy test Test cases with shock 36
Numerical results Nonuniform Meshes 37
Numerical results WENO limiter on unstructured meshes 38
Numerical results 2D Euler equation, HWENO limiter unstructured meshes 39
Numerical results 2D Euler quation, New HWENO limiter 40
Numerical results Lax problem: Euler equations with initial condition (ρ, v, p) = { (0.445, 0.698, 3.528) if x 0, (0.5, 0, 0.571) if x > 0. 41
Numerical results 42
Numerical results New HWENO limiter, P 1, P 2 and P 3 from left to right using KXRCF troubled-cell indicator. 43
Numerical results Blast wave problem: Euler equations with initial condition (1, 0, 1000) if 0 x < 0.1, (ρ, v, p) = (1, 0, 0.01) if 0.1 x < 0.9, (1, 0, 100) if 0.9 x 1. 44 First Prev Next Last Go Back Full Screen Close Quit
Numerical results 45
Numerical results New HWENO limiter, P 1, P 2 and P 3 from left to right using KXRCF troubled-cell indicator. 46
Numerical results Two-dimensional Euler equations The PDEs are ρ ρu ρu ρu ρv + 2 + p ρuv + E u(e + p) t x ρv ρuv ρv 2 + p v(e + p) y = 0. 47
Numerical results Double mach reflection problem The computational domain for this problem is [0, 4] [0, 1]. The reflecting wall lies at the bottom, starting from x = 1 6. Initially a right-moving Mach 10 shock is positioned at x = 1 6, y = 0 and makes a 60 angle with the x-axis. For the bottom boundary, the exact post-shock condition is imposed for the part from x = 0 to x = 1 6 and a reflective boundary condition is used for the rest. At the top boundary, the flow values are set to describe the exact motion of a Mach 10 shock. We compute the solution up to t = 0.2. 48
Numerical results Form top to bottom, WENO limiter (1920 480 cells), HWENO limiter (1920 480 cells) and New HWENO limiter (800 200 cells). 49
Numerical results Forward step problem A Mach 3 wind tunnel with a step. The wind tunnel is 1 length unit wide and 3 length units long. The step is 0.2 length units high and is located 0.6 length units from the left-hand end of the tunnel. The problem is initialized by a right-going Mach 3 flow. Reflective boundary conditions are applied along the wall of the tunnel and in/out flow boundary conditions are applied at the entrance/exit. We compute the solution up to t = 4. 50
Numerical results Form top to bottom, WENO limiter (480 160 cells), HWENO limiter (240 80 cells) and New HWENO limiter (240 80 cells). 51
Conclusions 4 Conclusions We have developed a new limiter for the RKDG methods solving hyperbolic conservation laws using finite volume high order WENO and HWENO reconstructions. First identify troubled cells by troubled cell indicator. Then reconstruct the polynomial solution inside the troubled cells by WENO type reconstruction using the cell averages and moments of neighboring cells, while maintaining the original cell averages of the troubled cells. Numerical results show that the methods are stable, accurate and robust in maintaining accuracy. Troubled cell indicator was used as discontinuous indicator for h-adaptive and r- adaptive methods (H. Zhu Q. JCP 2009, ACM 2013) and hybrid WENO methods (G. Li Q, JCP 2010, JSC 2012, ACM 2013 ). 52
Refereces 1. J. Qiu and C.-W. Shu: J. Comput. Phys., 193 (2004) 115-135. 2. J. Qiu and C.-W. Shu: Computers & Fluids, 34 (2005) 642-663. 3. J. Qiu and C.-W. Shu: SIAM J. Sci. Comput., 26 (2005), 907-929. 4. J. Qiu and C.-W. Shu: SIAM J. Sci. Comput., 27 (2005), 995-1013. 5. J. Zhu, J. Qiu, C.-W. Shu and M. Dumbser: J. Comput. Phys., 227 (2008) 4330-4353. 6. J. Zhu and J. Qiu: J. Sci. Comput., 39 (2009), 293-321. 7. H. Zhu and J. Qiu: J. Comput. Phys., 228 (2009), 6957-6976. 8. G. Li and J. Qiu: J. Comput. Phys., 229 (2010), 8105-8129. 9. J. Zhu and J. Qiu: J. Comput. Phys., 230 (2011), 4353-4375. 10. J. Zhu and J. Qiu: Commun.Comput. Phys., 11 (2012), 985-1005. 11. G. Li, C. Lu and J. Qiu: J. Sci. Comput., 51 (2012), 527-559. 12. J. Zhu and J. Qiu: J. Sci. Comput., 55 (2013), 606-644. 13. H. Zhu, Y. Cheng and J. Qiu: Adv. Appl. Math. Mech., 5 (2013), 365-390. 14. J. Zhu, X. Zhong, C.-W. Shu and J. Qiu: J. Comput. Phys., 248 (2013), 200-220. 15. H. Zhu and J. Qiu: Adv. Comput. Math., 39(2013), 445-463. 16. G. Li and J. Qiu: Adv. Comput. Math., 40 (2014), 747-772. 17. J. Zhu, X. Zhong, C.-W. Shu and J. Qiu: Runge-Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter, submitted to Commun.Comput. Phys. 53
THANK YOU! jxqiu@xmu.edu.cn http://ccam.xmu.edu.cn/teacher/jxqiu